Could in-homogeneous ODE has more than one particular solution $x_p$? I study ODE course at MIT open course, and the professor said several times "any particular solution would be fine".
So, Could in-homogeneous ODE has more than one particular solution $x_p$ ? If so I want an example.
If the particular solution is the system response to the input signal, how could be more than one response ?
Thanks for help.
 A: The general solution of an ODE includes arbitrary constant(s). For example in a first order ODE such as :
$$x'(t)+2x(t)=4t$$
the general solution is : $\quad x(t)=2t+C\:e^{-2t}$
where $C$ is any constant.
So, for example, $x_p=2t+3e^{-2t}$ is a solution of the ODE. That is a particular solution. 
A manner to express the general solution is :
$$x(t)=x_p+C\:e^{-2t} =(2t+3e^{-2t})+C\:e^{-2t}$$
which is equivalent than above because $(3+C)$ is also any constant.
Also $x_p=2t+5e^{-2t}$ is a solution of the ODE. That is a particular solution.
Also $x_p=2t$ is a solution of the ODE. That is a particular solution.
Also $x_p=2t-13e^{-2t}$ is a solution of the ODE. It is a particular solution. 
You see that they are many particular solutions.
Now, if we no longer consider the ODE alone, but the ODE with a given initial solution, this becomes more specific. For example, given the initial condition :
$$x(0)=3$$
Looks what happen if you take, for example, the particular solution $x_p=2t-13e^{-2t}$
$$x(t)=x_p+C\:e^{-2t} =(2t-13e^{-2t})+C\:e^{-2t}$$
$x(0)=3=(0-13e^{-0})+C\:e^{-0}=-13+C \quad\to\quad 3=-13+C \quad\to\quad C=16$
$x(t)=x_p+16\:e^{-2t} =(2t-13e^{-2t})+16e^{-2t}$
$$ x(t)= 2t+3e^{-2t}$$
Try to do it with another particular solution $x_p$. You will see that the result will be exactly the same : $x(t)= 2t+3e^{-2t}$. 
For A GIVEN INITIAL CONDITION, the result is always the same whatever the particular solution used.
In case of more complicated system (and more complicated ODE), if the particular solution is the system response to the input signal, there is only one response because the initial condition is given when the input signal is applied.
